Properties

Label 4560.2.a.bn.1.1
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +0.585786 q^{11} +0.585786 q^{13} -1.00000 q^{15} -2.82843 q^{17} -1.00000 q^{19} -1.41421 q^{21} +4.82843 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.07107 q^{29} +4.82843 q^{31} +0.585786 q^{33} +1.41421 q^{35} +6.24264 q^{37} +0.585786 q^{39} +9.89949 q^{41} -11.0711 q^{43} -1.00000 q^{45} +3.17157 q^{47} -5.00000 q^{49} -2.82843 q^{51} +8.48528 q^{53} -0.585786 q^{55} -1.00000 q^{57} +1.17157 q^{59} -1.65685 q^{61} -1.41421 q^{63} -0.585786 q^{65} +11.3137 q^{67} +4.82843 q^{69} +14.8284 q^{71} -11.6569 q^{73} +1.00000 q^{75} -0.828427 q^{77} +13.6569 q^{79} +1.00000 q^{81} +7.65685 q^{83} +2.82843 q^{85} -7.07107 q^{87} -12.2426 q^{89} -0.828427 q^{91} +4.82843 q^{93} +1.00000 q^{95} +11.8995 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} + 4 q^{33} + 4 q^{37} + 4 q^{39} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 10 q^{49} - 4 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} + 24 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{89} + 4 q^{91} + 4 q^{93} + 2 q^{95} + 4 q^{97} + 4 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.07107 −1.31306 −0.656532 0.754298i \(-0.727977\pi\)
−0.656532 + 0.754298i \(0.727977\pi\)
\(30\) 0 0
\(31\) 4.82843 0.867211 0.433606 0.901103i \(-0.357241\pi\)
0.433606 + 0.901103i \(0.357241\pi\)
\(32\) 0 0
\(33\) 0.585786 0.101972
\(34\) 0 0
\(35\) 1.41421 0.239046
\(36\) 0 0
\(37\) 6.24264 1.02628 0.513142 0.858304i \(-0.328481\pi\)
0.513142 + 0.858304i \(0.328481\pi\)
\(38\) 0 0
\(39\) 0.585786 0.0938009
\(40\) 0 0
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) −11.0711 −1.68832 −0.844161 0.536090i \(-0.819901\pi\)
−0.844161 + 0.536090i \(0.819901\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 3.17157 0.462621 0.231311 0.972880i \(-0.425699\pi\)
0.231311 + 0.972880i \(0.425699\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) 8.48528 1.16554 0.582772 0.812636i \(-0.301968\pi\)
0.582772 + 0.812636i \(0.301968\pi\)
\(54\) 0 0
\(55\) −0.585786 −0.0789874
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 1.17157 0.152526 0.0762629 0.997088i \(-0.475701\pi\)
0.0762629 + 0.997088i \(0.475701\pi\)
\(60\) 0 0
\(61\) −1.65685 −0.212138 −0.106069 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) −0.585786 −0.0726579
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) 4.82843 0.581274
\(70\) 0 0
\(71\) 14.8284 1.75981 0.879905 0.475149i \(-0.157606\pi\)
0.879905 + 0.475149i \(0.157606\pi\)
\(72\) 0 0
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −0.828427 −0.0944080
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.65685 0.840449 0.420224 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) −7.07107 −0.758098
\(88\) 0 0
\(89\) −12.2426 −1.29772 −0.648859 0.760909i \(-0.724753\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(90\) 0 0
\(91\) −0.828427 −0.0868428
\(92\) 0 0
\(93\) 4.82843 0.500685
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 11.8995 1.20821 0.604105 0.796904i \(-0.293531\pi\)
0.604105 + 0.796904i \(0.293531\pi\)
\(98\) 0 0
\(99\) 0.585786 0.0588738
\(100\) 0 0
\(101\) −0.828427 −0.0824316 −0.0412158 0.999150i \(-0.513123\pi\)
−0.0412158 + 0.999150i \(0.513123\pi\)
\(102\) 0 0
\(103\) −3.31371 −0.326509 −0.163255 0.986584i \(-0.552199\pi\)
−0.163255 + 0.986584i \(0.552199\pi\)
\(104\) 0 0
\(105\) 1.41421 0.138013
\(106\) 0 0
\(107\) 13.6569 1.32026 0.660129 0.751152i \(-0.270502\pi\)
0.660129 + 0.751152i \(0.270502\pi\)
\(108\) 0 0
\(109\) 8.14214 0.779875 0.389938 0.920841i \(-0.372497\pi\)
0.389938 + 0.920841i \(0.372497\pi\)
\(110\) 0 0
\(111\) 6.24264 0.592525
\(112\) 0 0
\(113\) −15.3137 −1.44059 −0.720296 0.693667i \(-0.755994\pi\)
−0.720296 + 0.693667i \(0.755994\pi\)
\(114\) 0 0
\(115\) −4.82843 −0.450253
\(116\) 0 0
\(117\) 0.585786 0.0541560
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) 0 0
\(123\) 9.89949 0.892607
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) −11.0711 −0.974753
\(130\) 0 0
\(131\) −6.72792 −0.587821 −0.293911 0.955833i \(-0.594957\pi\)
−0.293911 + 0.955833i \(0.594957\pi\)
\(132\) 0 0
\(133\) 1.41421 0.122628
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 15.6569 1.33766 0.668828 0.743417i \(-0.266796\pi\)
0.668828 + 0.743417i \(0.266796\pi\)
\(138\) 0 0
\(139\) 2.82843 0.239904 0.119952 0.992780i \(-0.461726\pi\)
0.119952 + 0.992780i \(0.461726\pi\)
\(140\) 0 0
\(141\) 3.17157 0.267095
\(142\) 0 0
\(143\) 0.343146 0.0286953
\(144\) 0 0
\(145\) 7.07107 0.587220
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) −4.34315 −0.355804 −0.177902 0.984048i \(-0.556931\pi\)
−0.177902 + 0.984048i \(0.556931\pi\)
\(150\) 0 0
\(151\) −0.828427 −0.0674164 −0.0337082 0.999432i \(-0.510732\pi\)
−0.0337082 + 0.999432i \(0.510732\pi\)
\(152\) 0 0
\(153\) −2.82843 −0.228665
\(154\) 0 0
\(155\) −4.82843 −0.387829
\(156\) 0 0
\(157\) −4.14214 −0.330578 −0.165289 0.986245i \(-0.552856\pi\)
−0.165289 + 0.986245i \(0.552856\pi\)
\(158\) 0 0
\(159\) 8.48528 0.672927
\(160\) 0 0
\(161\) −6.82843 −0.538155
\(162\) 0 0
\(163\) 23.5563 1.84508 0.922538 0.385907i \(-0.126111\pi\)
0.922538 + 0.385907i \(0.126111\pi\)
\(164\) 0 0
\(165\) −0.585786 −0.0456034
\(166\) 0 0
\(167\) 17.3137 1.33977 0.669887 0.742463i \(-0.266342\pi\)
0.669887 + 0.742463i \(0.266342\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −11.3137 −0.860165 −0.430083 0.902790i \(-0.641516\pi\)
−0.430083 + 0.902790i \(0.641516\pi\)
\(174\) 0 0
\(175\) −1.41421 −0.106904
\(176\) 0 0
\(177\) 1.17157 0.0880608
\(178\) 0 0
\(179\) 13.1716 0.984490 0.492245 0.870457i \(-0.336176\pi\)
0.492245 + 0.870457i \(0.336176\pi\)
\(180\) 0 0
\(181\) 20.1421 1.49715 0.748577 0.663048i \(-0.230738\pi\)
0.748577 + 0.663048i \(0.230738\pi\)
\(182\) 0 0
\(183\) −1.65685 −0.122478
\(184\) 0 0
\(185\) −6.24264 −0.458968
\(186\) 0 0
\(187\) −1.65685 −0.121161
\(188\) 0 0
\(189\) −1.41421 −0.102869
\(190\) 0 0
\(191\) 13.7574 0.995448 0.497724 0.867336i \(-0.334169\pi\)
0.497724 + 0.867336i \(0.334169\pi\)
\(192\) 0 0
\(193\) 13.0711 0.940876 0.470438 0.882433i \(-0.344096\pi\)
0.470438 + 0.882433i \(0.344096\pi\)
\(194\) 0 0
\(195\) −0.585786 −0.0419490
\(196\) 0 0
\(197\) 22.1421 1.57756 0.788781 0.614674i \(-0.210713\pi\)
0.788781 + 0.614674i \(0.210713\pi\)
\(198\) 0 0
\(199\) 9.17157 0.650156 0.325078 0.945687i \(-0.394610\pi\)
0.325078 + 0.945687i \(0.394610\pi\)
\(200\) 0 0
\(201\) 11.3137 0.798007
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) −9.89949 −0.691411
\(206\) 0 0
\(207\) 4.82843 0.335599
\(208\) 0 0
\(209\) −0.585786 −0.0405197
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 14.8284 1.01603
\(214\) 0 0
\(215\) 11.0711 0.755041
\(216\) 0 0
\(217\) −6.82843 −0.463544
\(218\) 0 0
\(219\) −11.6569 −0.787697
\(220\) 0 0
\(221\) −1.65685 −0.111452
\(222\) 0 0
\(223\) −11.3137 −0.757622 −0.378811 0.925474i \(-0.623667\pi\)
−0.378811 + 0.925474i \(0.623667\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −11.6569 −0.773693 −0.386846 0.922144i \(-0.626436\pi\)
−0.386846 + 0.922144i \(0.626436\pi\)
\(228\) 0 0
\(229\) −21.6569 −1.43113 −0.715563 0.698549i \(-0.753830\pi\)
−0.715563 + 0.698549i \(0.753830\pi\)
\(230\) 0 0
\(231\) −0.828427 −0.0545065
\(232\) 0 0
\(233\) 15.6569 1.02571 0.512857 0.858474i \(-0.328587\pi\)
0.512857 + 0.858474i \(0.328587\pi\)
\(234\) 0 0
\(235\) −3.17157 −0.206891
\(236\) 0 0
\(237\) 13.6569 0.887108
\(238\) 0 0
\(239\) 17.7574 1.14863 0.574314 0.818635i \(-0.305269\pi\)
0.574314 + 0.818635i \(0.305269\pi\)
\(240\) 0 0
\(241\) 12.3431 0.795092 0.397546 0.917582i \(-0.369862\pi\)
0.397546 + 0.917582i \(0.369862\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.00000 0.319438
\(246\) 0 0
\(247\) −0.585786 −0.0372727
\(248\) 0 0
\(249\) 7.65685 0.485233
\(250\) 0 0
\(251\) 5.75736 0.363401 0.181701 0.983354i \(-0.441840\pi\)
0.181701 + 0.983354i \(0.441840\pi\)
\(252\) 0 0
\(253\) 2.82843 0.177822
\(254\) 0 0
\(255\) 2.82843 0.177123
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) −8.82843 −0.548572
\(260\) 0 0
\(261\) −7.07107 −0.437688
\(262\) 0 0
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) −8.48528 −0.521247
\(266\) 0 0
\(267\) −12.2426 −0.749237
\(268\) 0 0
\(269\) 14.3848 0.877055 0.438528 0.898718i \(-0.355500\pi\)
0.438528 + 0.898718i \(0.355500\pi\)
\(270\) 0 0
\(271\) 2.82843 0.171815 0.0859074 0.996303i \(-0.472621\pi\)
0.0859074 + 0.996303i \(0.472621\pi\)
\(272\) 0 0
\(273\) −0.828427 −0.0501387
\(274\) 0 0
\(275\) 0.585786 0.0353243
\(276\) 0 0
\(277\) −24.6274 −1.47972 −0.739859 0.672762i \(-0.765108\pi\)
−0.739859 + 0.672762i \(0.765108\pi\)
\(278\) 0 0
\(279\) 4.82843 0.289070
\(280\) 0 0
\(281\) −30.3848 −1.81260 −0.906302 0.422631i \(-0.861107\pi\)
−0.906302 + 0.422631i \(0.861107\pi\)
\(282\) 0 0
\(283\) −7.75736 −0.461127 −0.230564 0.973057i \(-0.574057\pi\)
−0.230564 + 0.973057i \(0.574057\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −14.0000 −0.826394
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 11.8995 0.697561
\(292\) 0 0
\(293\) 1.65685 0.0967945 0.0483972 0.998828i \(-0.484589\pi\)
0.0483972 + 0.998828i \(0.484589\pi\)
\(294\) 0 0
\(295\) −1.17157 −0.0682116
\(296\) 0 0
\(297\) 0.585786 0.0339908
\(298\) 0 0
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) 15.6569 0.902446
\(302\) 0 0
\(303\) −0.828427 −0.0475919
\(304\) 0 0
\(305\) 1.65685 0.0948712
\(306\) 0 0
\(307\) −30.1421 −1.72030 −0.860151 0.510039i \(-0.829631\pi\)
−0.860151 + 0.510039i \(0.829631\pi\)
\(308\) 0 0
\(309\) −3.31371 −0.188510
\(310\) 0 0
\(311\) 23.2132 1.31630 0.658150 0.752887i \(-0.271339\pi\)
0.658150 + 0.752887i \(0.271339\pi\)
\(312\) 0 0
\(313\) 25.7990 1.45825 0.729123 0.684383i \(-0.239928\pi\)
0.729123 + 0.684383i \(0.239928\pi\)
\(314\) 0 0
\(315\) 1.41421 0.0796819
\(316\) 0 0
\(317\) 26.1421 1.46829 0.734144 0.678993i \(-0.237584\pi\)
0.734144 + 0.678993i \(0.237584\pi\)
\(318\) 0 0
\(319\) −4.14214 −0.231915
\(320\) 0 0
\(321\) 13.6569 0.762251
\(322\) 0 0
\(323\) 2.82843 0.157378
\(324\) 0 0
\(325\) 0.585786 0.0324936
\(326\) 0 0
\(327\) 8.14214 0.450261
\(328\) 0 0
\(329\) −4.48528 −0.247282
\(330\) 0 0
\(331\) 12.1421 0.667392 0.333696 0.942681i \(-0.391704\pi\)
0.333696 + 0.942681i \(0.391704\pi\)
\(332\) 0 0
\(333\) 6.24264 0.342095
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 17.7574 0.967305 0.483652 0.875260i \(-0.339310\pi\)
0.483652 + 0.875260i \(0.339310\pi\)
\(338\) 0 0
\(339\) −15.3137 −0.831726
\(340\) 0 0
\(341\) 2.82843 0.153168
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −4.82843 −0.259954
\(346\) 0 0
\(347\) −21.3137 −1.14418 −0.572090 0.820191i \(-0.693867\pi\)
−0.572090 + 0.820191i \(0.693867\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0.585786 0.0312670
\(352\) 0 0
\(353\) −6.68629 −0.355875 −0.177938 0.984042i \(-0.556943\pi\)
−0.177938 + 0.984042i \(0.556943\pi\)
\(354\) 0 0
\(355\) −14.8284 −0.787011
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −13.5563 −0.715477 −0.357738 0.933822i \(-0.616452\pi\)
−0.357738 + 0.933822i \(0.616452\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −10.6569 −0.559340
\(364\) 0 0
\(365\) 11.6569 0.610148
\(366\) 0 0
\(367\) −24.7279 −1.29079 −0.645394 0.763850i \(-0.723307\pi\)
−0.645394 + 0.763850i \(0.723307\pi\)
\(368\) 0 0
\(369\) 9.89949 0.515347
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −17.0711 −0.883906 −0.441953 0.897038i \(-0.645714\pi\)
−0.441953 + 0.897038i \(0.645714\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −4.14214 −0.213331
\(378\) 0 0
\(379\) −9.51472 −0.488738 −0.244369 0.969682i \(-0.578581\pi\)
−0.244369 + 0.969682i \(0.578581\pi\)
\(380\) 0 0
\(381\) 5.65685 0.289809
\(382\) 0 0
\(383\) 34.6274 1.76938 0.884689 0.466181i \(-0.154371\pi\)
0.884689 + 0.466181i \(0.154371\pi\)
\(384\) 0 0
\(385\) 0.828427 0.0422206
\(386\) 0 0
\(387\) −11.0711 −0.562774
\(388\) 0 0
\(389\) −34.9706 −1.77308 −0.886539 0.462654i \(-0.846897\pi\)
−0.886539 + 0.462654i \(0.846897\pi\)
\(390\) 0 0
\(391\) −13.6569 −0.690657
\(392\) 0 0
\(393\) −6.72792 −0.339379
\(394\) 0 0
\(395\) −13.6569 −0.687151
\(396\) 0 0
\(397\) 16.6274 0.834506 0.417253 0.908790i \(-0.362993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(398\) 0 0
\(399\) 1.41421 0.0707992
\(400\) 0 0
\(401\) 18.3848 0.918092 0.459046 0.888413i \(-0.348191\pi\)
0.459046 + 0.888413i \(0.348191\pi\)
\(402\) 0 0
\(403\) 2.82843 0.140894
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.65685 0.181264
\(408\) 0 0
\(409\) −4.82843 −0.238750 −0.119375 0.992849i \(-0.538089\pi\)
−0.119375 + 0.992849i \(0.538089\pi\)
\(410\) 0 0
\(411\) 15.6569 0.772296
\(412\) 0 0
\(413\) −1.65685 −0.0815285
\(414\) 0 0
\(415\) −7.65685 −0.375860
\(416\) 0 0
\(417\) 2.82843 0.138509
\(418\) 0 0
\(419\) 1.55635 0.0760326 0.0380163 0.999277i \(-0.487896\pi\)
0.0380163 + 0.999277i \(0.487896\pi\)
\(420\) 0 0
\(421\) −16.6274 −0.810371 −0.405185 0.914235i \(-0.632793\pi\)
−0.405185 + 0.914235i \(0.632793\pi\)
\(422\) 0 0
\(423\) 3.17157 0.154207
\(424\) 0 0
\(425\) −2.82843 −0.137199
\(426\) 0 0
\(427\) 2.34315 0.113393
\(428\) 0 0
\(429\) 0.343146 0.0165672
\(430\) 0 0
\(431\) −6.14214 −0.295856 −0.147928 0.988998i \(-0.547260\pi\)
−0.147928 + 0.988998i \(0.547260\pi\)
\(432\) 0 0
\(433\) −14.2426 −0.684458 −0.342229 0.939617i \(-0.611182\pi\)
−0.342229 + 0.939617i \(0.611182\pi\)
\(434\) 0 0
\(435\) 7.07107 0.339032
\(436\) 0 0
\(437\) −4.82843 −0.230975
\(438\) 0 0
\(439\) −4.68629 −0.223664 −0.111832 0.993727i \(-0.535672\pi\)
−0.111832 + 0.993727i \(0.535672\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) −23.4558 −1.11442 −0.557210 0.830371i \(-0.688128\pi\)
−0.557210 + 0.830371i \(0.688128\pi\)
\(444\) 0 0
\(445\) 12.2426 0.580357
\(446\) 0 0
\(447\) −4.34315 −0.205424
\(448\) 0 0
\(449\) −14.3848 −0.678860 −0.339430 0.940631i \(-0.610234\pi\)
−0.339430 + 0.940631i \(0.610234\pi\)
\(450\) 0 0
\(451\) 5.79899 0.273064
\(452\) 0 0
\(453\) −0.828427 −0.0389229
\(454\) 0 0
\(455\) 0.828427 0.0388373
\(456\) 0 0
\(457\) 16.1421 0.755097 0.377549 0.925990i \(-0.376767\pi\)
0.377549 + 0.925990i \(0.376767\pi\)
\(458\) 0 0
\(459\) −2.82843 −0.132020
\(460\) 0 0
\(461\) 6.68629 0.311412 0.155706 0.987803i \(-0.450235\pi\)
0.155706 + 0.987803i \(0.450235\pi\)
\(462\) 0 0
\(463\) −24.7279 −1.14920 −0.574602 0.818433i \(-0.694843\pi\)
−0.574602 + 0.818433i \(0.694843\pi\)
\(464\) 0 0
\(465\) −4.82843 −0.223913
\(466\) 0 0
\(467\) 16.8284 0.778727 0.389363 0.921084i \(-0.372695\pi\)
0.389363 + 0.921084i \(0.372695\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −4.14214 −0.190860
\(472\) 0 0
\(473\) −6.48528 −0.298194
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 8.48528 0.388514
\(478\) 0 0
\(479\) −5.75736 −0.263060 −0.131530 0.991312i \(-0.541989\pi\)
−0.131530 + 0.991312i \(0.541989\pi\)
\(480\) 0 0
\(481\) 3.65685 0.166738
\(482\) 0 0
\(483\) −6.82843 −0.310704
\(484\) 0 0
\(485\) −11.8995 −0.540328
\(486\) 0 0
\(487\) −23.7990 −1.07844 −0.539218 0.842166i \(-0.681280\pi\)
−0.539218 + 0.842166i \(0.681280\pi\)
\(488\) 0 0
\(489\) 23.5563 1.06525
\(490\) 0 0
\(491\) 12.3848 0.558917 0.279459 0.960158i \(-0.409845\pi\)
0.279459 + 0.960158i \(0.409845\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −0.585786 −0.0263291
\(496\) 0 0
\(497\) −20.9706 −0.940658
\(498\) 0 0
\(499\) −1.85786 −0.0831694 −0.0415847 0.999135i \(-0.513241\pi\)
−0.0415847 + 0.999135i \(0.513241\pi\)
\(500\) 0 0
\(501\) 17.3137 0.773519
\(502\) 0 0
\(503\) 6.68629 0.298127 0.149064 0.988828i \(-0.452374\pi\)
0.149064 + 0.988828i \(0.452374\pi\)
\(504\) 0 0
\(505\) 0.828427 0.0368645
\(506\) 0 0
\(507\) −12.6569 −0.562111
\(508\) 0 0
\(509\) −4.92893 −0.218471 −0.109236 0.994016i \(-0.534840\pi\)
−0.109236 + 0.994016i \(0.534840\pi\)
\(510\) 0 0
\(511\) 16.4853 0.729266
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 3.31371 0.146019
\(516\) 0 0
\(517\) 1.85786 0.0817088
\(518\) 0 0
\(519\) −11.3137 −0.496617
\(520\) 0 0
\(521\) 7.55635 0.331050 0.165525 0.986206i \(-0.447068\pi\)
0.165525 + 0.986206i \(0.447068\pi\)
\(522\) 0 0
\(523\) −23.7990 −1.04066 −0.520329 0.853966i \(-0.674191\pi\)
−0.520329 + 0.853966i \(0.674191\pi\)
\(524\) 0 0
\(525\) −1.41421 −0.0617213
\(526\) 0 0
\(527\) −13.6569 −0.594902
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) 1.17157 0.0508419
\(532\) 0 0
\(533\) 5.79899 0.251182
\(534\) 0 0
\(535\) −13.6569 −0.590437
\(536\) 0 0
\(537\) 13.1716 0.568395
\(538\) 0 0
\(539\) −2.92893 −0.126158
\(540\) 0 0
\(541\) −13.3137 −0.572401 −0.286201 0.958170i \(-0.592392\pi\)
−0.286201 + 0.958170i \(0.592392\pi\)
\(542\) 0 0
\(543\) 20.1421 0.864382
\(544\) 0 0
\(545\) −8.14214 −0.348771
\(546\) 0 0
\(547\) 8.48528 0.362804 0.181402 0.983409i \(-0.441936\pi\)
0.181402 + 0.983409i \(0.441936\pi\)
\(548\) 0 0
\(549\) −1.65685 −0.0707128
\(550\) 0 0
\(551\) 7.07107 0.301238
\(552\) 0 0
\(553\) −19.3137 −0.821302
\(554\) 0 0
\(555\) −6.24264 −0.264985
\(556\) 0 0
\(557\) 12.3431 0.522996 0.261498 0.965204i \(-0.415784\pi\)
0.261498 + 0.965204i \(0.415784\pi\)
\(558\) 0 0
\(559\) −6.48528 −0.274298
\(560\) 0 0
\(561\) −1.65685 −0.0699524
\(562\) 0 0
\(563\) −11.6569 −0.491278 −0.245639 0.969361i \(-0.578998\pi\)
−0.245639 + 0.969361i \(0.578998\pi\)
\(564\) 0 0
\(565\) 15.3137 0.644253
\(566\) 0 0
\(567\) −1.41421 −0.0593914
\(568\) 0 0
\(569\) 34.1838 1.43306 0.716529 0.697557i \(-0.245730\pi\)
0.716529 + 0.697557i \(0.245730\pi\)
\(570\) 0 0
\(571\) −26.1421 −1.09401 −0.547007 0.837128i \(-0.684233\pi\)
−0.547007 + 0.837128i \(0.684233\pi\)
\(572\) 0 0
\(573\) 13.7574 0.574722
\(574\) 0 0
\(575\) 4.82843 0.201359
\(576\) 0 0
\(577\) −26.4853 −1.10260 −0.551298 0.834308i \(-0.685867\pi\)
−0.551298 + 0.834308i \(0.685867\pi\)
\(578\) 0 0
\(579\) 13.0711 0.543215
\(580\) 0 0
\(581\) −10.8284 −0.449239
\(582\) 0 0
\(583\) 4.97056 0.205860
\(584\) 0 0
\(585\) −0.585786 −0.0242193
\(586\) 0 0
\(587\) 23.4558 0.968126 0.484063 0.875033i \(-0.339160\pi\)
0.484063 + 0.875033i \(0.339160\pi\)
\(588\) 0 0
\(589\) −4.82843 −0.198952
\(590\) 0 0
\(591\) 22.1421 0.910806
\(592\) 0 0
\(593\) −0.627417 −0.0257649 −0.0128825 0.999917i \(-0.504101\pi\)
−0.0128825 + 0.999917i \(0.504101\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 9.17157 0.375367
\(598\) 0 0
\(599\) 16.9706 0.693398 0.346699 0.937976i \(-0.387302\pi\)
0.346699 + 0.937976i \(0.387302\pi\)
\(600\) 0 0
\(601\) 45.7990 1.86818 0.934090 0.357038i \(-0.116213\pi\)
0.934090 + 0.357038i \(0.116213\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 0 0
\(605\) 10.6569 0.433263
\(606\) 0 0
\(607\) 17.4558 0.708511 0.354255 0.935149i \(-0.384734\pi\)
0.354255 + 0.935149i \(0.384734\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 1.85786 0.0751611
\(612\) 0 0
\(613\) −7.17157 −0.289657 −0.144829 0.989457i \(-0.546263\pi\)
−0.144829 + 0.989457i \(0.546263\pi\)
\(614\) 0 0
\(615\) −9.89949 −0.399186
\(616\) 0 0
\(617\) 47.1127 1.89669 0.948343 0.317247i \(-0.102758\pi\)
0.948343 + 0.317247i \(0.102758\pi\)
\(618\) 0 0
\(619\) −20.4853 −0.823373 −0.411686 0.911326i \(-0.635060\pi\)
−0.411686 + 0.911326i \(0.635060\pi\)
\(620\) 0 0
\(621\) 4.82843 0.193758
\(622\) 0 0
\(623\) 17.3137 0.693659
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.585786 −0.0233941
\(628\) 0 0
\(629\) −17.6569 −0.704025
\(630\) 0 0
\(631\) −20.2843 −0.807504 −0.403752 0.914868i \(-0.632294\pi\)
−0.403752 + 0.914868i \(0.632294\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −5.65685 −0.224485
\(636\) 0 0
\(637\) −2.92893 −0.116049
\(638\) 0 0
\(639\) 14.8284 0.586604
\(640\) 0 0
\(641\) 31.0711 1.22723 0.613617 0.789604i \(-0.289714\pi\)
0.613617 + 0.789604i \(0.289714\pi\)
\(642\) 0 0
\(643\) 43.0711 1.69856 0.849279 0.527945i \(-0.177037\pi\)
0.849279 + 0.527945i \(0.177037\pi\)
\(644\) 0 0
\(645\) 11.0711 0.435923
\(646\) 0 0
\(647\) −2.68629 −0.105609 −0.0528045 0.998605i \(-0.516816\pi\)
−0.0528045 + 0.998605i \(0.516816\pi\)
\(648\) 0 0
\(649\) 0.686292 0.0269393
\(650\) 0 0
\(651\) −6.82843 −0.267627
\(652\) 0 0
\(653\) −18.1421 −0.709957 −0.354978 0.934875i \(-0.615512\pi\)
−0.354978 + 0.934875i \(0.615512\pi\)
\(654\) 0 0
\(655\) 6.72792 0.262882
\(656\) 0 0
\(657\) −11.6569 −0.454777
\(658\) 0 0
\(659\) 4.97056 0.193626 0.0968128 0.995303i \(-0.469135\pi\)
0.0968128 + 0.995303i \(0.469135\pi\)
\(660\) 0 0
\(661\) 13.1127 0.510025 0.255012 0.966938i \(-0.417920\pi\)
0.255012 + 0.966938i \(0.417920\pi\)
\(662\) 0 0
\(663\) −1.65685 −0.0643469
\(664\) 0 0
\(665\) −1.41421 −0.0548408
\(666\) 0 0
\(667\) −34.1421 −1.32199
\(668\) 0 0
\(669\) −11.3137 −0.437413
\(670\) 0 0
\(671\) −0.970563 −0.0374682
\(672\) 0 0
\(673\) 5.75736 0.221930 0.110965 0.993824i \(-0.464606\pi\)
0.110965 + 0.993824i \(0.464606\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 34.8284 1.33857 0.669283 0.743008i \(-0.266602\pi\)
0.669283 + 0.743008i \(0.266602\pi\)
\(678\) 0 0
\(679\) −16.8284 −0.645816
\(680\) 0 0
\(681\) −11.6569 −0.446692
\(682\) 0 0
\(683\) −12.6863 −0.485427 −0.242714 0.970098i \(-0.578038\pi\)
−0.242714 + 0.970098i \(0.578038\pi\)
\(684\) 0 0
\(685\) −15.6569 −0.598218
\(686\) 0 0
\(687\) −21.6569 −0.826261
\(688\) 0 0
\(689\) 4.97056 0.189363
\(690\) 0 0
\(691\) −47.7990 −1.81836 −0.909180 0.416404i \(-0.863290\pi\)
−0.909180 + 0.416404i \(0.863290\pi\)
\(692\) 0 0
\(693\) −0.828427 −0.0314693
\(694\) 0 0
\(695\) −2.82843 −0.107288
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 15.6569 0.592197
\(700\) 0 0
\(701\) −10.9706 −0.414352 −0.207176 0.978304i \(-0.566427\pi\)
−0.207176 + 0.978304i \(0.566427\pi\)
\(702\) 0 0
\(703\) −6.24264 −0.235446
\(704\) 0 0
\(705\) −3.17157 −0.119448
\(706\) 0 0
\(707\) 1.17157 0.0440615
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 13.6569 0.512172
\(712\) 0 0
\(713\) 23.3137 0.873105
\(714\) 0 0
\(715\) −0.343146 −0.0128329
\(716\) 0 0
\(717\) 17.7574 0.663161
\(718\) 0 0
\(719\) −10.2426 −0.381986 −0.190993 0.981591i \(-0.561171\pi\)
−0.190993 + 0.981591i \(0.561171\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 0 0
\(723\) 12.3431 0.459047
\(724\) 0 0
\(725\) −7.07107 −0.262613
\(726\) 0 0
\(727\) −24.2426 −0.899110 −0.449555 0.893253i \(-0.648417\pi\)
−0.449555 + 0.893253i \(0.648417\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 31.3137 1.15818
\(732\) 0 0
\(733\) −10.6863 −0.394707 −0.197354 0.980332i \(-0.563235\pi\)
−0.197354 + 0.980332i \(0.563235\pi\)
\(734\) 0 0
\(735\) 5.00000 0.184428
\(736\) 0 0
\(737\) 6.62742 0.244124
\(738\) 0 0
\(739\) −7.31371 −0.269039 −0.134520 0.990911i \(-0.542949\pi\)
−0.134520 + 0.990911i \(0.542949\pi\)
\(740\) 0 0
\(741\) −0.585786 −0.0215194
\(742\) 0 0
\(743\) −41.2548 −1.51349 −0.756747 0.653708i \(-0.773212\pi\)
−0.756747 + 0.653708i \(0.773212\pi\)
\(744\) 0 0
\(745\) 4.34315 0.159121
\(746\) 0 0
\(747\) 7.65685 0.280150
\(748\) 0 0
\(749\) −19.3137 −0.705708
\(750\) 0 0
\(751\) 44.1421 1.61077 0.805385 0.592752i \(-0.201959\pi\)
0.805385 + 0.592752i \(0.201959\pi\)
\(752\) 0 0
\(753\) 5.75736 0.209810
\(754\) 0 0
\(755\) 0.828427 0.0301496
\(756\) 0 0
\(757\) −26.4853 −0.962624 −0.481312 0.876549i \(-0.659840\pi\)
−0.481312 + 0.876549i \(0.659840\pi\)
\(758\) 0 0
\(759\) 2.82843 0.102665
\(760\) 0 0
\(761\) −39.6569 −1.43756 −0.718780 0.695238i \(-0.755299\pi\)
−0.718780 + 0.695238i \(0.755299\pi\)
\(762\) 0 0
\(763\) −11.5147 −0.416861
\(764\) 0 0
\(765\) 2.82843 0.102262
\(766\) 0 0
\(767\) 0.686292 0.0247805
\(768\) 0 0
\(769\) −34.2843 −1.23632 −0.618161 0.786051i \(-0.712122\pi\)
−0.618161 + 0.786051i \(0.712122\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) 5.45584 0.196233 0.0981165 0.995175i \(-0.468718\pi\)
0.0981165 + 0.995175i \(0.468718\pi\)
\(774\) 0 0
\(775\) 4.82843 0.173442
\(776\) 0 0
\(777\) −8.82843 −0.316718
\(778\) 0 0
\(779\) −9.89949 −0.354686
\(780\) 0 0
\(781\) 8.68629 0.310820
\(782\) 0 0
\(783\) −7.07107 −0.252699
\(784\) 0 0
\(785\) 4.14214 0.147839
\(786\) 0 0
\(787\) 6.82843 0.243407 0.121704 0.992566i \(-0.461164\pi\)
0.121704 + 0.992566i \(0.461164\pi\)
\(788\) 0 0
\(789\) 2.00000 0.0712019
\(790\) 0 0
\(791\) 21.6569 0.770029
\(792\) 0 0
\(793\) −0.970563 −0.0344657
\(794\) 0 0
\(795\) −8.48528 −0.300942
\(796\) 0 0
\(797\) 4.68629 0.165997 0.0829985 0.996550i \(-0.473550\pi\)
0.0829985 + 0.996550i \(0.473550\pi\)
\(798\) 0 0
\(799\) −8.97056 −0.317356
\(800\) 0 0
\(801\) −12.2426 −0.432572
\(802\) 0 0
\(803\) −6.82843 −0.240970
\(804\) 0 0
\(805\) 6.82843 0.240670
\(806\) 0 0
\(807\) 14.3848 0.506368
\(808\) 0 0
\(809\) −53.3137 −1.87441 −0.937205 0.348779i \(-0.886596\pi\)
−0.937205 + 0.348779i \(0.886596\pi\)
\(810\) 0 0
\(811\) −52.9706 −1.86005 −0.930024 0.367499i \(-0.880214\pi\)
−0.930024 + 0.367499i \(0.880214\pi\)
\(812\) 0 0
\(813\) 2.82843 0.0991973
\(814\) 0 0
\(815\) −23.5563 −0.825143
\(816\) 0 0
\(817\) 11.0711 0.387328
\(818\) 0 0
\(819\) −0.828427 −0.0289476
\(820\) 0 0
\(821\) −21.7990 −0.760790 −0.380395 0.924824i \(-0.624212\pi\)
−0.380395 + 0.924824i \(0.624212\pi\)
\(822\) 0 0
\(823\) 50.8701 1.77322 0.886609 0.462519i \(-0.153054\pi\)
0.886609 + 0.462519i \(0.153054\pi\)
\(824\) 0 0
\(825\) 0.585786 0.0203945
\(826\) 0 0
\(827\) −7.65685 −0.266255 −0.133127 0.991099i \(-0.542502\pi\)
−0.133127 + 0.991099i \(0.542502\pi\)
\(828\) 0 0
\(829\) −0.544156 −0.0188993 −0.00944966 0.999955i \(-0.503008\pi\)
−0.00944966 + 0.999955i \(0.503008\pi\)
\(830\) 0 0
\(831\) −24.6274 −0.854316
\(832\) 0 0
\(833\) 14.1421 0.489996
\(834\) 0 0
\(835\) −17.3137 −0.599166
\(836\) 0 0
\(837\) 4.82843 0.166895
\(838\) 0 0
\(839\) −34.1421 −1.17872 −0.589359 0.807871i \(-0.700620\pi\)
−0.589359 + 0.807871i \(0.700620\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) −30.3848 −1.04651
\(844\) 0 0
\(845\) 12.6569 0.435409
\(846\) 0 0
\(847\) 15.0711 0.517848
\(848\) 0 0
\(849\) −7.75736 −0.266232
\(850\) 0 0
\(851\) 30.1421 1.03326
\(852\) 0 0
\(853\) 3.17157 0.108593 0.0542963 0.998525i \(-0.482708\pi\)
0.0542963 + 0.998525i \(0.482708\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) −25.4558 −0.869555 −0.434778 0.900538i \(-0.643173\pi\)
−0.434778 + 0.900538i \(0.643173\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) −14.0000 −0.477119
\(862\) 0 0
\(863\) 57.2548 1.94898 0.974489 0.224437i \(-0.0720543\pi\)
0.974489 + 0.224437i \(0.0720543\pi\)
\(864\) 0 0
\(865\) 11.3137 0.384678
\(866\) 0 0
\(867\) −9.00000 −0.305656
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 6.62742 0.224561
\(872\) 0 0
\(873\) 11.8995 0.402737
\(874\) 0 0
\(875\) 1.41421 0.0478091
\(876\) 0 0
\(877\) 2.04163 0.0689410 0.0344705 0.999406i \(-0.489026\pi\)
0.0344705 + 0.999406i \(0.489026\pi\)
\(878\) 0 0
\(879\) 1.65685 0.0558843
\(880\) 0 0
\(881\) −17.5147 −0.590086 −0.295043 0.955484i \(-0.595334\pi\)
−0.295043 + 0.955484i \(0.595334\pi\)
\(882\) 0 0
\(883\) 44.2426 1.48888 0.744442 0.667687i \(-0.232716\pi\)
0.744442 + 0.667687i \(0.232716\pi\)
\(884\) 0 0
\(885\) −1.17157 −0.0393820
\(886\) 0 0
\(887\) 10.3431 0.347289 0.173644 0.984808i \(-0.444446\pi\)
0.173644 + 0.984808i \(0.444446\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0.585786 0.0196246
\(892\) 0 0
\(893\) −3.17157 −0.106133
\(894\) 0 0
\(895\) −13.1716 −0.440277
\(896\) 0 0
\(897\) 2.82843 0.0944384
\(898\) 0 0
\(899\) −34.1421 −1.13870
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 15.6569 0.521027
\(904\) 0 0
\(905\) −20.1421 −0.669547
\(906\) 0 0
\(907\) 20.7696 0.689642 0.344821 0.938669i \(-0.387940\pi\)
0.344821 + 0.938669i \(0.387940\pi\)
\(908\) 0 0
\(909\) −0.828427 −0.0274772
\(910\) 0 0
\(911\) −34.6274 −1.14726 −0.573629 0.819115i \(-0.694465\pi\)
−0.573629 + 0.819115i \(0.694465\pi\)
\(912\) 0 0
\(913\) 4.48528 0.148441
\(914\) 0 0
\(915\) 1.65685 0.0547739
\(916\) 0 0
\(917\) 9.51472 0.314204
\(918\) 0 0
\(919\) 51.5980 1.70206 0.851030 0.525117i \(-0.175978\pi\)
0.851030 + 0.525117i \(0.175978\pi\)
\(920\) 0 0
\(921\) −30.1421 −0.993217
\(922\) 0 0
\(923\) 8.68629 0.285913
\(924\) 0 0
\(925\) 6.24264 0.205257
\(926\) 0 0
\(927\) −3.31371 −0.108836
\(928\) 0 0
\(929\) −14.2010 −0.465920 −0.232960 0.972486i \(-0.574841\pi\)
−0.232960 + 0.972486i \(0.574841\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 0 0
\(933\) 23.2132 0.759966
\(934\) 0 0
\(935\) 1.65685 0.0541849
\(936\) 0 0
\(937\) −29.7990 −0.973491 −0.486745 0.873544i \(-0.661816\pi\)
−0.486745 + 0.873544i \(0.661816\pi\)
\(938\) 0 0
\(939\) 25.7990 0.841918
\(940\) 0 0
\(941\) 7.75736 0.252883 0.126441 0.991974i \(-0.459644\pi\)
0.126441 + 0.991974i \(0.459644\pi\)
\(942\) 0 0
\(943\) 47.7990 1.55655
\(944\) 0 0
\(945\) 1.41421 0.0460044
\(946\) 0 0
\(947\) −38.2843 −1.24407 −0.622036 0.782989i \(-0.713694\pi\)
−0.622036 + 0.782989i \(0.713694\pi\)
\(948\) 0 0
\(949\) −6.82843 −0.221660
\(950\) 0 0
\(951\) 26.1421 0.847717
\(952\) 0 0
\(953\) 36.9706 1.19759 0.598797 0.800901i \(-0.295646\pi\)
0.598797 + 0.800901i \(0.295646\pi\)
\(954\) 0 0
\(955\) −13.7574 −0.445178
\(956\) 0 0
\(957\) −4.14214 −0.133896
\(958\) 0 0
\(959\) −22.1421 −0.715007
\(960\) 0 0
\(961\) −7.68629 −0.247945
\(962\) 0 0
\(963\) 13.6569 0.440086
\(964\) 0 0
\(965\) −13.0711 −0.420773
\(966\) 0 0
\(967\) 17.4142 0.560003 0.280002 0.960000i \(-0.409665\pi\)
0.280002 + 0.960000i \(0.409665\pi\)
\(968\) 0 0
\(969\) 2.82843 0.0908622
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) 0.585786 0.0187602
\(976\) 0 0
\(977\) 3.51472 0.112446 0.0562229 0.998418i \(-0.482094\pi\)
0.0562229 + 0.998418i \(0.482094\pi\)
\(978\) 0 0
\(979\) −7.17157 −0.229204
\(980\) 0 0
\(981\) 8.14214 0.259958
\(982\) 0 0
\(983\) 51.9411 1.65666 0.828332 0.560237i \(-0.189290\pi\)
0.828332 + 0.560237i \(0.189290\pi\)
\(984\) 0 0
\(985\) −22.1421 −0.705507
\(986\) 0 0
\(987\) −4.48528 −0.142768
\(988\) 0 0
\(989\) −53.4558 −1.69980
\(990\) 0 0
\(991\) 21.6569 0.687953 0.343976 0.938978i \(-0.388226\pi\)
0.343976 + 0.938978i \(0.388226\pi\)
\(992\) 0 0
\(993\) 12.1421 0.385319
\(994\) 0 0
\(995\) −9.17157 −0.290758
\(996\) 0 0
\(997\) −23.4558 −0.742854 −0.371427 0.928462i \(-0.621131\pi\)
−0.371427 + 0.928462i \(0.621131\pi\)
\(998\) 0 0
\(999\) 6.24264 0.197508
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4560.2.a.bn.1.1 2
4.3 odd 2 2280.2.a.k.1.2 2
12.11 even 2 6840.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.k.1.2 2 4.3 odd 2
4560.2.a.bn.1.1 2 1.1 even 1 trivial
6840.2.a.bb.1.2 2 12.11 even 2